An introductory example: the uniform static field

This chapter discusses some physical effects related to two simple metrics: the RIndler metric and the uniform static field. The purpose is to illustrate the methods by applying them in an exact calculation which is not too taxing. The Christoffel symbols and curvature tensors are obtained, and some example geodesics are calculated. The force experienced by a fisherman fishing in the RIndler metric is calculated.

Parallel transport and geodesics

The mathematics of parallel transport and of affine and metric geodesics is presented. The geodesic equation is obtained in several different ways, bringing out its role both as a geometric statement and as an equation of motion. The Euler-Lagrange method to find metric geodesics, and hence Christoffel symbols, is explained. The role of conserved quantities is discussed. Killing’s equation and Killing vectors are introduced. Fermi-Walker transport (the non-rotating freely falling cabin) is defined and discussed. Gravitational redshift is calculated, first in general and then in specific cases.

Tensors

Tensors and tensor algebra are presented. The concept of a tensor is defined in two ways: as something which yields a scalar from a set of vectors, and as something whose components transform a given way. The meaning and use of these definitions is expounded carefully, along with examples. The action of the metric and its inverse (index lowering and raising) is derived. The relation between geodesic coordinates and Christoffel symbols is obtained. The difference between partial differentiation and covariant differentiation is explained at length. The tensor density and Hodge dual are briefly introduced.

Schwarzschild–Droste solution

The spherically symmetric vacuum solution to the Einstein field equation (Schwarzschild-Droste solution) is derived and associated physical phenomena derived and explained. It is shown how to obtain the Christoffel symbols by the Euler-Lagrange method, and hence the metric for the general spherically symmetric vacuum. Equations for general orbits are presented, and their solution for radial motion and for circular motion. Geodetic (de Sitter) precession is calculated exactly for circular orbits. The null geodesics (photon worldlines) are obtained, and the gravitational redshift. Emission from an accretion disc is calculated.

Hubble Expansion as an Einstein Curvature

Hubble expansion may be considered as a velocity per photon travel time rather than as velocity or redshift per distance. Dimensionally, this is an acceleration and will have an associated curvature of space under general relativity. This paper explores this theoretical curvature as an extension to the spacetime manifold of general relativity, generating a modified solution with three additional non-zero Christoffel symbols, and a reformulated Ricci tensor and curvature. The observational consequences of this reformulation were compared with the ΛCDM model for luminosity distance using the extensive type Ia supernovae (SNe Ia) data with redshift corrected to the CMB, and for angular diameter distance with the recent baryonic acoustic oscillation (BAO) data. For the SNe Ia data, the modified GR and ΛCDM models differed by −0.15+0.11μB mag. over zcmb=0.01−1.3, with overall weighted RMS errors of ±0.136μB mag for modified GR and ±0.151μB mag for ΛCDM espectively. The BAO measures spanned a range z=0.106−2.36, with weighted RMS errors of ±0.034 Mpc with H0=67.6±0.25 for the modified GR model, and ±0.085 Mpc with H0=70.0±0.25 for the ΛCDM model. The derived GR metric for this new solution describes both the SNe Ia and the BAO observations with comparable accuracy to ΛCDM without requiring the inclusion of dark matter or w’-corrected dark energy.

About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho ei is found the expressions of vector projections (ΔA) i and (rot A) i , where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA) i and (rot A) i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA) i and (rot A) i as an example are made in cases of spherical and cylindrical coordinate systems.

Numerical Methods to Compute the Coriolis Matrix and Christoffel Symbols for Rigid-body Systems

This article presents methods to efficiently compute the Coriolis matrix and underlying Christoffel symbols (of the first kind) for tree-structure rigid-body systems. The algorithms can be executed purely numerically, without requiring partial derivatives as in unscalable symbolic techniques. The computations share a recursive structure in common with classical methods such as the Composite-Rigid-Body Algorithm and are of the lowest possible order: $O(Nd)$ for the Coriolis matrix and $O(Nd^2)$ for the Christoffel symbols, where $N$ is the number of bodies and $d$ is the depth of the kinematic tree. Implementation in C/C++ shows computation times on the order of 10-20 $\mu$s for the Coriolis matrix and 40-120 $\mu$s for the Christoffel symbols on systems with 20 degrees of freedom. The results demonstrate feasibility for the adoption of these algorithms within high-rate (>1kHz) loops for model-based control applications.

Tensors

In this chapter we develop the concept of tensors, their meaning, and how they arise from vectors. Emphasis is placed on tensor transformations, covariance between coordinate systems, and relation to the metric. The concept of metric connection and the Christoffel symbols is introduced in three dimensions via the easily visualizable idea of parallel transport. Derivatives and intergrals in covariant form are discussed. The first two chapters are designed to familiarize the reader with the language that is the bread and butter of the general theory of relativity and other higher geometric theories.

Didactic demonstration of 3gdl robot dynamics and partitioned control for trajectory tracking

The article presents the analysis of a robot with three degrees of freedom to follow trajectories through a partitioned control. Which is made up of two revolute and one prismatic joint where the end effector is located, that allows it to move correctly in its work area. This robot has a different structure from those most studied and analyzed by current literature, therefore it presents an opportunity to be used as a didactic resource, due to the structure, the degrees of freedom and the affinity of the models used by the students. The analysis consists of the use of the DH rule for the assignment of frames and referential axes, centers of mass, dynamic model by Jacobian and Christoffel symbols, inverse kinematic model, variables such as friction, gravitational and friction compensation, ending in a model in "Simulink" capable of following trajectories from the partitioned control law.